Let $$\left( {\left\{ {p,\,q} \right\},\, * } \right)$$ be a semi group where $$p * p = q$$. Show that: (a) $$p * q = q * p,$$, and (b) $$q * q = q$$

(a) Show that the formula $$\left[ {\left( { \sim p \vee Q} \right) \Rightarrow \left( {q \Rightarrow p} \right)} \right]$$ is not a tautology.

(b) Let $$A$$ be a tautology and $$B$$ be any other formula. Prove that $$\left( {A \vee B} \right)$$ is a tautology.

Show that proposition $$C$$ is a logical consequence of the formula $$A \wedge \left( {A \to \left( {B \vee C} \right) \wedge \left( {B \to \sim A} \right)} \right)$$ using truth tables.

Uses Modus ponens $$\left( {A,\,\,A \to B\,|\,\, = B} \right)$$ or resolution to show that the following set is inconsistent:

(1) $$Q\left( x \right) \to P\left( x \right)V \sim R\left( a \right)$$
(2) $$R\left( a \right) \vee \sim Q\left( a \right)$$
(3) $$Q\left( a \right)$$
(4) $$ \sim P\left( y \right)$$
where $$x$$ and $$y$$ are universally quantifies variables, $$a$$ is a constant and $$P, Q, R$$ are monadic predicates.